Put the eggplant in a colander and sprinkle evenly with 2 teaspoons salt. Weight the eggplant down, and let drain for about 1 hour, stirring the pieces after 30 minutes. Rinse the eggplant and press it very dry with paper towels.

While the eggplant is draining, heat 3 tablespoons oil in a 1--to- 12-inch heavy skillet over medium heat. Add the onions, and salt and pepper them lightly. Cook for 2 minutes, stirring so they are thoroughly coated with oil. Cover, turn the heat down to the lowest simmer, and cook slowly until the onions are meltingly tender, 35-40 minutes. Stir from time to time to make sure the onions don't burn. When they are very soft, remove the lid, raise the heat to high, and brown them to a rich caramel gold. Stir frequently with a wooden spoon to redistribute the syrupy juices. If necessary, turn the heat down a bit to prevent the onions from sticking and burning. When the onions are thick and jamlike, stir in the thyme or marjoram. Adjust the seasoning and transfer the mixture to a very large bowl.

Lightly rinse out the skillet and dry it, add 2 tablespoons fresh oil, and turn the heat to high. Add the mushrooms and cook, stirring frequently, until they release some juice, about 5 minutes. Sprinkle with salt and pepper, add the garlic, and continue sautéing, lifting and turning often until all the liquid has evaporated and the mushrooms are golden brown, about 7 minutes. Add the mushrooms to the onions.

Wipe out the skillet and in it heat the remaining 3 tablespoons oil over medium-high heat until hot but not smoking. Add the eggplant, in batches if necessary, and fry until tender and lightly browned on both sides. Add more oil to the skillet if needed, but always make sure the oil is very hot before adding the eggplant--this will prevent the eggplant from absorbing too much oil. Transfer the eggplant to the onions and mushrooms in the bowl.

Preheat the oven to 350 degrees F.

Prepare the kasha. In a medium bowl, beat the egg with a fork. Stir in the kasha and mix until each grain is thoroughly coated with egg. Heat the broth to simmering. In a heavy skillet with high sides or a wide, heavy saucepan, toast the kasha over medium heat, turning and breaking up the kasha constantly until the egg begins to dry and the grains separate, about 3 minutes. Add the hot broth and salt and pepper to taste, then cover and simmer over very low heat until tender and all the liquid is absorbed, 10-15 minutes.

Meanwhile, bring 2 quarts of water and 1 1/2 teaspoons salt to a rolling boil. Add the noodles and cook until tender but still firm to the bite. Drain and stir into the vegetables.

In a lightly greased 3-quart shallow casserole, combine the kasha with the other ingredients. Adjust the seasoning to taste. If the mixture seems dry, add schmaltz or dot with butter or margarine as needed. Bake just until heated through. Sprinkle with the scallions or chives and the parsley, and serve hot.